The date field (which is deprecated and not a standard anymore according to W3C) takes the date in the format you wrote: "yyyy-MM-dd". Carbon::now() outputs a full datetime value, like "2016-03-23 16:52:00.000".
You can use
Carbon::now()->toDateString();
which will output "2016-03-23".
But since you use a datetime field in your database, you'll need to use a datetime-local field.